In the text, it says:
Consider smooth functions on a manifold $X$: $f: X \to \mathbb{R}$, at a particular $x \in X$, $f$ is either regular or $df_x = 0$.
So I am not certain here: if $df_x \neq 0$, then all we know here is $df_x$ is injective. Hence how can we conclude that $df$ us surjective?
The image of $d_xf$ is a subspace of a one dimensional vector space: it is either zero or the whole thing.